Lattice Bose polarons at strong coupling and quantum criticality

In the present manuscript, the authors investigate the problem of an impurity particle immersed in an interacting Bose gas in a square lattice.

To describe the Bose medium, they use the quantum Gutzwiller method previously introduced by some of the authors to investigate the Bose-Hubbard model. Considering short range interactions between the impurity and the bath particles, the authors derive the relevant vertices in terms of the Bogoliubov operators describing the collective excitations of the medium obtained using the above mentioned method.

Then, they introduce a diagrammatic theory for the polaron self energy where the impurity-medium excitation scattering is treated in a ladder approximation, which goes beyond the recent perturbative theory introduced by some of the authors in Ref. [27].
In particular, the ladder diagrams capture the presence of bound states between the impurity and a medium particle.

Focusing on the case where the medium state is in the vicinity of the O(2) critical point between Mott-insulator and superfluid phase, the authors use their theory to calculate and analyze the impurity spectral function across the transition.
In addition, they present comparisons between the results of their diagrammatic approach and the ground state obtained from Monte Carlo simulations, and find a remarkable agreement.

I found the paper clearly written and the results rather well explained.
The results presented are original and seem correct. In my opinion, the present manuscript may be suitable publication once the authors have considered my remarks below.

Please find below my comments:

- It is maybe a naive question, but could the authors comment on the reason they use of a finite size system for the calculation based on 'quantum Gutzwiller' method? Is it not possible to consider the homogeneous and infinite case (which, I believe has been used to obtain Figure 1)?

- The definition of the vertices in Eq. 5) involve sums over n. I presume that these sums are truncated at some n_max, could the authors comment on how they choose the value of such n_max?

- Eqs. 7) and 10) contain the same products of operators and are represented by the same diagrams in Fig. 2a). Similarly Eqs. 6) and 11) are made of identical products of operators, but the corresponding diagrams in Figure 2a) differ, is this normal or is there a typo somewhere?


- In Section 4.3, the authors compare the results of their Gutzwiller diagrammatic approach with Monte Carlo calculations. In particular, the authors explain that in order to compare the regime of strong impurity-bath interactions, they need to adjust the density to a non-integer filling density (and hence outside the Mott phase) for the Gutzwiller method.
I wonder if a comparison could be done in the Mott phase if the MC simulations were performed with (M+1) or (M-1) bosons instead of M bosons?

- The authors briefly mention that the Gutzwiller method is inaccurate for the determination of the O2 critical point. Could the authors elaborate on the limitations of the method and in particular on how this could affect the polaron properties?

- In the appendix A2, the authors refer to particle-particle and hole-hole processes that are equal and opposite in the Mott-phase. Do these correspond to the case with $\lambda\lambda'=11 $ and $ 22$? Since the figure 8 shows many lines, it may be beneficial for the reader to be explicit.

- I find equation A12) a bit confusing. I could not find the definition of $D^{(0)}$ nearby, and perhaps the sum over $k$ should instead be over $\bf q$?

- It seems that a factor $U_{IB}$ may be missing in the second term in Eq. A13)?

- It seems that the footnote p.8 and the sentence line 174-175 repeat themselves.

- line 490 typo: 'has been have been'

Publish (meets expectations and criteria for this Journal)

The manuscript by R. Alhyder et al. discusses the properties of mobile impurities interacting strongly with a bosonic lattice gas, which is tuned across the SF-MI quantum phase transition.

The paper is very well-written, the diagrammatic and QMC calculations are challenging but explained in detail, the graphs are illustrative, and the results appear correct. Furthermore, the considered set-up may be engineered in currently available experiments. For all these reasons, I can certainly recommend the publication of this manuscript in Scipost Physics.

Here below I list some remarks and comments, that the authors could implement in a revised version of the manuscript:

1 - the acronym QGW for "Quantum Gutzwiller" is somehow strange, given that "Gutzwiller" is a single word; i.e., what is the letter "W" standing for?

2 - in the abstract, and also on line 617: "resuming" ---> "resumming"

3 - the word "Mottness" may be familiar to some specialists, but is certainly not of general use. As such it sounds weird, specially when used in the Abstract. I suggest to rephrase that sentence, removing this word, or at least explaining it better.

4 - in Fig.1, it should be clarified that the two lowest panels compose Fig.1b. Else, call 1b the left panel, and 1c the right one.

5 - regarding Fig.1b(left), it is unclear to me why the lowest branch is called "particle", and the upper one "hole". How do the two excitations differ?

6 - in Eq.(1), I believe that a "minus" sign is missing in front of the first $\sum$ sign (i.e., it should be $-t$, rather than $t$)

7 - on line 102 there is a wrong link: Fig. 3.1.1(a) ---> Fig. 1(a)

8 - in Sec. 3.1 (or in Sec. 4 at the latest), I would specify somewhere that calculations are performed with $\lambda = 0,1,2$ at most. As of now, this is only stated in one of the Appendices.

9 - lines 142-144: could the authors explain more in detail with $\mu'(t)<0$ means "hole SF", while $\mu'(t)>0$ means "particle SF"?

10 - line 145: interfering ---> interacting

11 - I noticed that more than once Figures appeared much before the page where they were referenced, making it hard for the reader to follow the narration (in such a long and complex paper). Figures should appear on the page where they are referred to, or after that.

12 - $M$ is undefined in Eq.(14). Only much later the reader learns that $M$ is the total number of sites (i.e., the system volume)

13 - on line 264: "considering a superfluid fraction" ---> "considering a condensate fraction"

14 - in Fig. 4, the red line plays a key role, but its explicit formula is hidden inside Ref.[27]. I suggest to quote the explicit formula giving the red line (2nd order perturbation result) in this paper as well.

15 - line 377: "is a finite-size" ---> "are a finite-size"

16 - line 432: "superfluid" ---> "condensate"

17 - The relation between Figs. 6 and 7, and a good part of discussion in Sec. 4.3 were unclear to me. I understand that QMC works in the canonical (at fixed number of particle), while QGW is in the grand-canonical, this is clear. But why Fig.6a is so different from 7a, when the filling has changed by just 1%? (for example, why do the two red lines differ so much?)
Moreover, it is difficult to compare the various panels in figures 6 and 7.
Maybe the two figures may be joined in a single one, containing a grid of 4*2 panels? (if so, then the same values of U_{IB}/U may be used in panels b and c of Figs. 6 and 7)

18 - regarding the orthogonality catastrophe (OC): in the caption of Fig.7 it is said that the ladder resummation "remedies" the OC, and similarly on line 589 the word "resolves" is used. Notice however that in Ref. [64] a very different message is passed: in the context of usual Bose polarons, the authors of [64] showed that the OC must be present when the Bose bath becomes ideal (i.e., non interacting), and that a ladder treatment is clearly not able to recover it (while a GPE approach can).
Notice furthermore that QMC calculations are normally performed with ~100 particles, and become inaccurate exactly in the OC limit, where the size of the dressing cloud explodes. For a discussion of finite-size effects in QMC, see for example the discussion around the vertical arrows in Fig. 4 of [N. Yegovtsev et al., Phys. Rev. A 110, 023310 (2024)]

19 - lines 609-610: shouldn't all differences between canonical and grand-canonical disappear in the thermodynamic limit?

20 - line 627: "as have been analyzed" ---> "as analyzed"

21 - some references are duplicated: for example [7] and [83], or [20] and [66]

22 - check the spelling of the initials of the second author of Ref. [63]

23 - check Ref. [60]: "and O. U."?

24 - Eq. (A.1): clarify how is $H_B$ related to $H$ given in Eq.(1). Similarly, it is unclear how Eq. (2) relates to Eq. (1)

25 - caption of Fig.8: "evaluate a" ---> "evaluated the vertices at a"

Publish (easily meets expectations and criteria for this Journal; among top 50%)